Abstract

Nonreciprocal devices that permit wave transmission in only one
direction are indispensible in many fields of science including,
e.g., electronics, optics, acoustics, and thermodynamics.
Manipulating phonons using such nonreciprocal devices may have a
range of applications such as phonon diodes, transistors,
switches, etc. One way of achieving nonreciprocal phononic devices
is to use materials with strong nonlinear response to phonons.
However, it is not easy to obtain the required strong mechanical
nonlinearity, especially for few-phonon situations. Here, we
present a general mechanism to amplify nonlinearity using
PT-symmetric structures, and show that an on-chip
micro-scale phonon diode can be fabricated using a
PT-symmetric mechanical system, in which a lossy
mechanical-resonator with very weak mechanical nonlinearity is
coupled to a mechanical resonator with mechanical gain but no
mechanical nonlinearity. When this coupled system transits from
the PT-symmetric regime to the
broken-PT-symmetric regime, the mechanical
nonlinearity is transferred from the lossy resonator to the one
with gain, and the effective nonlinearity of the system is
significantly enhanced. This enhanced mechanical nonlinearity is
almost lossless because of the gain-loss balance induced by the
PT-symmetric structure. Such an enhanced lossless
mechanical nonlinearity is then used to control the direction of
phonon propagation, and can greatly decrease (by over three orders
of magnitude) the threshold of the input-field intensity necessary
to observe the unidirectional phonon transport. We propose an
experimentally realizable lossless low-threshold phonon diode of
this type. Our study opens up new perspectives for constructing
on-chip few-phonon devices and hybrid phonon-photon components.

pacs:

Owing to recent progress in nanotechnology and materials science,
nano- and
micro-mechanics (1); (2); (3); (4); (5); (6); (7)
have emerged as subjects of great interest due to their potential
use in demonstrating macroscopic quantum phenomena, and possible
applications in precision measurements, detecting gravitational
waves, building filters, signal amplification, as well as switches
and logic gates. In particular, on-chip single- or few-phonon
devices are ideal candidates for hybrid quantum information
processing, due to the ability of phonons to interact and rapidly
switch between optical fields and microwave
fields (8); (9).
Fabrication of high-frequency mechanical
resonators (10), demonstration of coherent
phonon coupling between nanomechanical
resonators (11), ground-state
cooling (12); (13), optomechanics
(in microtoroids (14); (15),
microspheres (16); (17); (18),
microdisks (19); (20); (21),
microring (22), photonic
crystals (11), doubly- or singly-clamped
cantilevers (23); (24), and
membranes (25)) have opened new
directions (5) and provided new tools to
control and manipulate phonons in on-chip devices. One possible
obstacle to further develop this field is the ability to control
the flow of phonons, allowing transport in one direction but not
the opposite direction (26), i.e.,
nonreciprocal phonon transport. There have been several attempts
to fabricate nonreciprocal devices for
phonons (27); (28); (29); (30); (31); (32),
but these are almost exclusively based on asymmetric linear
structures which indeed cannot break Lorentz reciprocity: a static
linear structure cannot break reciprocity (27).
These proposed linear structures do obey the
reflection-transmission reciprocity and thus cannot be considered
as “phonon diodes”. Diode-like behavior was observed in these
linear acoustic structures because the input-output channels were
not properly switched (27).

Nonreciprocal phonon transmission inevitably requires
magneto-acoustic materials, strong nonlinearity, or a
time-dependent modulation of the parameters of a structure.
Although already demonstrated in optics (33),
the time-dependent modulation of acoustic parameters of a phononic
structure has not been probed yet. Magneto-acoustic materials
require high magnetic fields to operate and have been
studied (34); however, a magnetic-free
nonreciprocal device is critical for building on-chip and
small-scale phononic processors and circuits. Nonlinearity-based
nonreciprocity seems to be the most viable approach for creating
micro- or nano-scale nonreciprocal devices for controlling and
manipulating phonons.

Recently, there have been several reports on nonlinear mechanical
structures and
materials (35); (36); (37); (38).
However, the weak nonlinearity of those acoustic/phononic
materials hinders progress in this direction due to the high input
powers required to observe the nonlinear
effects (39); (40). In order to circumvent
this problem, coupling a weakly nonlinear structure to an
auxiliary system, such as a quantum bit (41),
has been proposed to engineer effective giant mechanical
nonlinearities.

In our proposed mechanical PT symmetric system, a
lossy mechanical resonator (passive resonator) which has a weak
mechanical nonlinearity is coupled to a mechanical resonator with
mechanical gain (active resonator) that balances the loss of the
passive resonator. The active resonator here works as a dynamical
amplifier. In the vicinity of the PT-phase transition,
the weak nonlinearity is first distributed between the
mechanically-coupled resonators and then significantly enhanced
due to the localization of the mechanical supermodes in the active
resonator. In this way, the effective nonlinear Kerr coefficient
is increased by over three orders of magnitude. This strong
nonlinearity, localized in the active resonator, blocks the phonon
transport from the active resonator to the lossy resonator but
permits the transport in the opposite direction.

For the experimental realization of the proposed
nonlinearity-based phonon diode, we provide a system in which a
mechanical beam with weak mechanical nonlinearity is coupled to
another mechanical beam with gain. We show that this micro-scale
system can be switched from a bidirectional transport regime to a
unidirectional transport regime, and vice versa, by properly
adjusting the detuning between the mechanical frequency of the
resonators and the frequency of the driving phononic field, or by
varying the amplitude of the input phononic field.

The system we consider here consists of two mechanical resonators,
one of which has mechanical loss (passive resonator) and weak
nonlinearity, and the other has mechanical gain (active resonator)
but no nonlinearity (see Fig. 1). The
mechanical coupling between the resonators is linear and it gives
rise to the mechanical supermodes b± with complex
eigenfrequencies

ω±=Ω±−iΓ±,

(1)

given by

ω±=Ω0−iχ±β.

(2)

Here Ω0 is the mechanical frequency of the solitary
mechanical resonators (i.e., both resonators are degenerate),

χ=(Γl−Γg)/2,

(3)

β=√g2mm−Γ2,

(4)

Γ=(Γl+Γg)/2,

(5)

where Γl and Γg denote, respectively, the damping
rate of the lossy mechanical resonator and the gain rate of the
active mechanical resonator, and gmm is the coupling strength
between the mechanical modes. When Γ≤gmm, the system
is in the PT-symmetric regime, and the supermodes are
non-degenerate with

Ω±=Ω0±β

(6)

and have the same damping rate χ (see Figs. 3a and 3b). However, when Γ>gmm, the
system is in the broken-PT-symmetric regime, the
supermodes are frequency-degenerate with Ω±=Ω0
(see Figs. 3a and
3b) and have
different damping rates

Γ±=χ∓iβ.

(7)

At Γ=gmm, the
two supermodes are degenerate with the same damping rate,
indicating a transition between the PT-symmetric regime and the
broken-PT-symmetry regime. This point is generally referred to as
the PT-transition point. It is seen that the two
supermodes will be lossless in the PT-symmetric regime
if the gain and loss are well-balanced, such that
Γl=Γg.

Figure 1: (Color online) Schematic diagram of the
proposed PT-symmetric mechanical system. The
PT-symmetric mechanical system has a linear mechanical
coupling between a passive mechanical resonator (having mechanical
loss and very weak mechanical nonlinearity) and an active
mechanical resonator (having mechanical gain but no nonlinearity).
Here blin and bgin are the input fields to the passive
and active resonators, respectively, and blout and
bgout are the output fields, respectively, leaving the
passive and active resonators. b1 and b2 denote the movable
resonators.

Let us assume that the passive resonator is made from a nonlinear
acoustic material (35) with a small nonlinear
Kerr coefficient μ. This nonlinearity mediates a cross-Kerr
interaction between the two mechanical supermodes, which leads to
the effective nonlinear coefficients μ′b and μ′s, in the
broken- and unbroken-PT regimes (69):

μ′b=μΓ2g2mm(Γ2−g2mm)2,μ′s=μg4mm(Γ2−g2mm)2.

(8)

Clearly, the effective nonlinear coefficients are significantly
enhanced in the vicinity of the phase transition point
Γ=gmm. Moreover, if the gain and loss are well-balanced,
i.e., Γl=Γg, the supermodes become almost lossless.
This observation is one of the key contributions of this paper.
Namely, operating the system of two coupled mechanical resonators
in the vicinity of the phase transition point will significantly
enhance the existing very weak nonlinearity with an extremely
small loss rate.

Figure 2: (Color online) Enhancement of
mechanical nonlinearity in a PT-symmetric mechanical system. The
coupling between two mechanical resonators creates two mechanical
supermodes symmetrically distributed between the resonators, and
hence both supermodes experience the weak nonlinearity of the
passive resonator. In the vicinity of the PT-phase transition,
which takes place when the coupling strength between the
resonators equals to the total loss in the system, the mechanical
nonlinearity is significantly enhanced due to localization of the
mechanical supermodes in the active mechanical
resonator.

Using the parameter values of μ/Ω0=10−5,
Γl/Ω0=0.55×10−3, and
Γg/Ω0=0.45×10−3, we show in Fig. 3 the evolution of the
eigenfrequencies of the system and of the nonlinear coefficient as
a function of gmm/Γ. The transition from the broken- to
the unbroken-PT symmetric regime and vice versa, as
the mechanical coupling strength is varied, is seen in Fig.
3a and 3b and it is reflected in the
bifurcations of the supermode frequencies and damping rates.
Moreover, the enhancement of the nonlinearity in the vicinity of
the PT-phase transition point is seen in Fig. 3c. We find that the
nonlinear coefficient is enhanced by more than three orders of
magnitude in the vicinity of the transition point.

More interestingly, in the broken-PT regime, the
mechanical energy of the coupled system is localized in the active
resonator, which leads to a nonlinear mechanical mode with strong
self-Kerr nonlinearity localized in the active mechanical
resonator. This can be interpreted intuitively as follows. The
initial weak mechanical nonlinearity is transferred from the
passive resonator to the active resonator and it is enhanced by
field localization in the broken-PT regime. Owing to
the presence of the mechanical gain, the active resonator then
enjoys an almost lossless mechanical mode with a giant
nonlinearity (see Fig. 2).

Finally, we would like to consider how the mechanical nonlinearity
will affect the PT-symmetric structure of the system.
Generally speaking, a strong nonlinearity will shift the
transition point of a PT-symmetric system or even
destroy the PT symmetry of such a
system (70). However, in our case, we
start from a system in which a gain resonator is coupled to a
lossy resonator with very weak Kerr nonlinearity, and thus we can
omit the shift of the PT-transition point induced by
such a weak nonlinearity. Although we generate a strong
nonlinearity in the vicinity of the PT-transition
point, this is an effective nonlinearity induced in the supermode
picture and thus will not affect the supermodes and the
PT-transition point of the system.

Figure 3: (Color online) Amplification of mechanical
nonlinearity via PT-symmetry breaking. (a) Effective
damping rates and (b) frequencies of the mechanical supermodes as
functions of the normalized mechanical coupling strength
gmm/Γ. (c) The effective nonlinear coefficients μ′b
in the PT-breaking regime and μ′s in the
PT-symmetric regime. The PT-phase
transition takes place at gmm=Γ. In the vicinity of this
transition point, the nonlinear coefficients μ′b and μ′s
are enhanced by more than three orders of magnitude (more than
35 dB increase compared to the baseline).

Here we investigate the effect of the enhanced mechanical
nonlinearity on the phonon transport in the coupled system. We
find that the localized strong mechanical nonlinearity leads to
unidirectional phonon transport from the passive resonator to the
active resonator and blocks phonon transport in the opposite
direction (i.e., phonon transport from the active to the passive
resonator is prevented). The transport is almost lossless due to
the gain-loss balance of the system. When this system is operated
in the vicinity of the PT-phase transition point, the
unidirectional phonon transport is possible within a region given
by (69)

δ∈[g2mmΩ0Ω20+χ2,g2mmΩ0−√3χ],

(9)

where

δ=Ω0−Ωd

(10)

is the detuning between the input (driving) field frequency
Ωd and the resonance frequency Ω0 of the
mechanical resonators. Additionally, in order to observe the
unidirectional phonon transport, the amplitude of the input field
should satisfy

|εd|2∈⎡⎣2(δ2+g2mm)39μ′bδ3,2(δ2+g2mm)39μ′bg2mmδ⎤⎦,

(11)

implying that the intensity of the input field required for
unidirectional transport is inversely proportional to the strength
of the mechanical nonlinearity μ′b. Since the strength of the
mechanical nonlinearity can be enhanced by more than three orders
of magnitude by breaking the PT symmetry, the
threshold of the input-field intensity for observing
unidirectional phonon transport can be decreased by at least three
orders of magnitude, allowing a low-threshold phonon diode
operation.

To show unidirectional phonon transport in the
broken-PT regime, let us first fix the amplitude of
the input field and vary the detuning δ. We compare the
amplitude transmittance

tl→g=bgout/blin

(12)

and

tg→l=blout/bgin.

(13)

The former, tl→g, denotes the transmission from
the passive to the active resonator, that is, the system is driven
by a phononic input field blin of frequency Ωd
at the passive resonator side and the output bgout is
measured at the active resonator side. However, the latter,
tg→l, denotes the amplitude transmittance from the
active resonator to the passive resonator when the system is
driven by the field bgin of frequency Ωd at the
active resonator side and the output blout is measured
at the passive side. The nonlinearity in the system manifests as a
bistability and hysteresis in the power transmittance,

Tg→l=∣∣tg→l∣∣2

(14)

and

Tl→g=∣∣tl→g∣∣2,

(15)

obtained as the detuning δ is up-scanned from smaller to
larger detuning and down-scanned from larger to smaller detuning
(see Fig. 4a).

We find that during the down-scan, both of the transmittances
Tl→g and Tg→l stay at the lower
branch with values close to zero until δ/Ω0=0.5×10−3, after which they bifurcate from each other only slightly
and then jump to the stable points at the upper branch of their
respective trajectories (see Fig. 4a). Further decreasing the detuning leads to an
increase in Tl→g, but a decrease in
Tg→l. This implies that there is no unidirectional
phonon transport with the parameter values used in the numerical
simulations. Instead, when the detuning is below a critical value,
the phonon transport is bidirectional; whereas when it is above
that critical value there is no phonon transport.

During the up-scan, however, after a short stay on the stable
state, i.e., a regime in which there is no bistability and
hysteresis in the transmittance, (during which Tl→g decreases and Tg→l increases with growing
detuning), both of the transmittances follow the upper branches of
their trajectories, during which a linear increase in
Tg→l and a slow-rate decrease in Tl→g are observed (see Fig. 4a). This behavior continues until
δ/Ω0∼2.5×10−3 for Tg→l,
where it jumps to the lower branch of its trajectory, and becomes
zero as the detuning is increased (see Fig. 4a). This implies that phonon
transport from the active mechanical resonator to the passive one
is prevented if the detuning is set to δ/Ω0>2.5×10−3. The transmittance Tl→g stays at its
upper branch with a value close to one until δ/Ω0∼3×10−3, where it jumps to its lower branch and becomes
zero. Thus, for δ/Ω0>3×10−3, phonon
transport from the passive to the active resonator is prevented.
Clearly, in the detuning region 2.5×10−3<δ/Ω0<3×10−3, the transmittance
Tl→g is close to one whereas Tg→l
is close to zero in this detuning region phonon transport from the
active mechanical resonator to the passive one is forbidden,
whereas phonon transport from the passive mechanical resonator to
active one is allowed with almost no loss. Thus, we conclude that
phonon transmission is non-reciprocal in this detuning region, and
the rectification is ∼30 dB within the nonreciprocal
transport region (see Fig. 4a). For detuning values smaller than the lower bound of
this region, phonon transport is bidirectional. For detuning
values larger than the upper bound of the region, phonon transport
is not possible.

Note that our phonon diode should work only when the disturbance
and perturbation of the system parameters are not too strong. In
fact, within the unidirectional phonon transport window shown in
Fig. 4a, the
transmittance Tl→g has two different branches of
metastable values. When we increase the detuning δ within
this unidirectional phonon transport window, Tl→g
will stay in the upper stable branch if we do not severely disturb
the system and the phonon diode should operate properly. However,
if the disturbance is too strong, Tl→g will jump
from the upper branch to the lower branch and stay in this stable
lower branch, without rectification.

Alternatively, we can fix the detuning and vary the amplitude of
the input field to show the nonlinearity-induced bistability and
hysteresis. A nonreciprocal phonon transport region is seen when
the amplitude of the input field is up-scanned (see Fig. 4b). The nonreciprocal transport
region disappears when the amplitude of the input field is
down-scanned. Within the nonreciprocal transport region, when the
input is varied at fixed detuning (see Fig. 4b), the rectification is
∼30 dB. Similarly, in this case, due to the metastability
of the transmittance Tl→g, the disturbance-induced
perturbation of the system parameters may not be too strong
otherwise our design of phonon diode will be invalid.

Figure 4: (Color online) Unidirectional phonon
transport by PT-symmetry breaking. (a) Unidirectional
phonon transport when the detuning δ is varied. The
transmittance from the active to passive mechanical resonator
Tg→l (red dash-dotted curve), and from the passive
to the active mechanical resonator Tl→g (blue
solid curve) versus the detuning δ=Ω0−Ωd shows
a strong bistability and hysteresis effect. The transmittance
functions evolve along different trajectories for increasing and
decreasing detuning due to the nonlinearity-induced bistability. A
unidirectional phonon-transport region (melon-colored shaded
region) appears only when the detuning δ is up-scanned from
smaller to larger detunings. Within this regime, the rectification
is ∼30 dB. (b) Unidirectional photon transport when the
amplitude of the input field is varied at fixed detuning
δ/Ω0=2.75×10−3. Within the unidirectional
transport region (melon-colored shaded region), rectification is
∼30 dB.

The unidirectional phonon transport enabled by the
PT-breaking-induced strong mechanical nonlinearity can
be used to fabricate lossless phonon diodes in on-chip systems.
This may have many applications, such as single-phonon transistors
and routers, on-chip quantum switches, and information-processing
components. One possible way to realize the proposed phonon diode
is to use coupled beams and cantilevers (see Fig. 5a). Phonon lasing, and hence an active
mechanical resonator, has been experimentally realized in an
electromechanical beam (71).
Elastically-coupled nano beams and cantilevers, by which the
mechanical supermodes can be generated, have also been shown in
various
experiments (72); (73); (74); (75),
in which the two mechanical resonators can be independently
driven (73). Thus our proposal is within the
reach of current experimental techniques of
nano-micro-electromechanical systems.

Let us now consider the design of the phonon diode system shown in
Fig. 5 in which a lossy vibrating
beam with damping rate Γl and a weak Kerr
nonlinearity (35) of strength μ is
elastically coupled to another vibrating beam with gain
Γg(71). The frequencies of the two
beams are both Ω0 and the mechanical coupling strength is
gmm.

Figure 5: (Color online) Schematic diagram of the
phonon diode system with two mechanical beams in which a beam with
weak mechanical nonlinearity is electrically or elastically
coupled to another beam with mechanical gain. The insets show the
finite-element-method (FEM) simulation by Comsol for the
mechanical modes.

In Fig. 6, we present the numerical
results performed with the system parameters: Ω0=600 kHz,
Γl=33 kHz, Γg=30 kHz, δ=1.65 kHz, μ=5.7
kHz, and gmm=1 kHz. Here, we fix the detuning δ and
change the amplitude of the input field. There is a 50 dB
background noise which includes the combined effect of the thermal
noise on the mechanical resonators, the electrical noises induced
by the measurement apparatus and other possible sources of noise.
The results shown in Fig. 6 for the
phonon diode agree well with the general model discussed in the
previous section. When the amplitude of the input field is
increased, it is clearly seen that there is a nonreciprocal region
in which phonon transport from the active beam to the passive beam
is almost completely suppressed (see Fig. 6b(ii)), but phonon transport from the passive beam to the
active beam is allowed (see Fig. 6a(ii)). A rectification ratio of about 30 dB is
obtained. When the amplitude of the phonon excitation is larger
than the upper bound of the unidirectional phonon transport
region, the transport is bidirectional. In this case, the phonons
can freely move from the active beam to the passive beam and vice
versa (see Figs. 6a(i) and 6b(i)). Finally, for amplitudes of the phonon
excitation smaller than the lower bound of the region, no phonon
transport can take place between the resonators (see
Figs. 6a(iii) and 6b(iii)). These are the result of hysteresis (see
Fig. 4b) caused by the
strong mechanical nonlinearity.

Figure 6: (Color online) Numerical results
demonstrating unidirectional phonon transport in a
PT-symmetric mechanical system in the
broken-PT phase. (a) Power spectrum obtained at the
output of the active beam without mechanical nonlinearity when the
phonon excitation (input) is at the passive beam with weak
nonlinearity. (b) Power spectrum obtained at the output of the
passive beam when the phonon excitation (input) is at the active
beam. When the intensity of the phonon excitation is within the
region bounded by Eq. (11), phonon transport is unidirectional. Transport from
the passive to the active resonator is allowed [see a(ii)], but
the transport from the active to the passive resonator is
prevented [b(ii)]. The rectification is about 30 dB. If the
intensity of the phonon excitation is larger than the upper bound
of the unidirectional transport region, phonon transport is
bidirectional [a(i) and b(i)]. Phonon transport is not allowed in
either of the directions [a(iii) and b(iii)] if the intensity of
the phonon excitation is smaller than the lower bound of the
region given in Eq. (11).

We have proposed a method to generate ultra-strong mechanical
nonlinearity with a very low-loss rate using a
PT-symmetric mechanical structure in which a
mechanical resonator with gain but no nonlinearity is coupled to a
lossy (i.e., passive mechanical loss and no gain) mechanical
resonator with very weak nonlinearity. We have showed that the
weak mechanical nonlinearity is redistributed in the supermodes of
the coupled mechanical system and is enhanced (by more than three
orders of magnitude) when the mechanical PT system
enters the broken-PT regime. Moreover, owing to the
presence of the mechanical gain in one of the resonators to
compensate the mechanical loss of the other resonator, the
effective mechanical damping rate is decreased in the
PT-symmetric system. Using experimentally accessible
parameter values, we identified the regimes where unidirectional
phonon transport is possible from the passive to active resonator
but not in the opposite direction. We then proposed an
experimentally-realizable system where a mechanical beam with
passive loss and weak nonlinearity is coupled to another beam
which acts like an active mechanical resonator. A possible
bottleneck for this design to achieve a phonon diode operated in
ambient condition is whether the mechanical gain observed with the
mechanical beams in a controlled environment and at low
temperatures (36) could also be obtained in
ambient-temperature conditions. A possible way to overcome this
problem, and to realize phonon diodes in ambient conditions, is to
use a hybrid system composed of a gain optomechanical resonator
and an nonlinear electrically-driven mechanical
beam (35), where the coupling between them is
achieved via the evanescent optical field of the optomechanical
resonator (76). The mechanical gain of
the optomechanical resonators can be provided at ambient
conditions by, e.g., the optomechanical dynamical instability in
the blue detuning regime (77), which has been
demonstrated in optomechanical resonators in various
experiments (78). Since creating
strongly-nonlinear mechanical or acoustic materials remains
challenging, we believe that the proposed system and the developed
approach provide a suitable platform for investigating nonlinear
phononics and can be used as a building block to design more
complex hybrid optomechanical or electromechanical information
processors. We envision that PT mechanical systems
will open a new route for designing functional phononic systems
with nonreciprocal phonon responses.

ACKNOWLEDGMENTS

JZ is supported by the NSFC under Grant Nos. 61174084, 61134008.
YXL is supported by the NSFC under Grant Nos. 10975080, 61025022,
91321208. YXL and JZ are supported by the National Basic Research
Program of China (973 Program) under Grant No. 2014CB921401, the
Tsinghua University Initiative Scientific Research Program, and
the Tsinghua National Laboratory for Information Science and
Technology (TNList) Cross-discipline Foundation. LY and SKO are
supported by ARO grant No. W911NF-12-1-0026 and the NSFC under
Grant No. 61328502. F.N. is supported by the RIKEN iTHES Project,
MURI Center for Dynamic Magneto-Optics via the AFOSR award number
FA9550-14-1-0040, and Grant-in-Aid for Scientific Research (A).

Appendix A Nonlinearity enhancement by broken PT symmetry

In order to prove the enhancement of mechanical nonlinearity in
the broken-PT-symmetric regime, let us consider a
system of two coupled mechanical resonators, in which one of the
resonators has mechanical gain (active resonator) and thus a
positive damping rate Γg and the second mechanical
resonator has a passive mechanical loss (passive resonator) with
loss rate Γl. The resonators have the same mechanical
frequency Ω0, and the annihilation operators for their
mechanical modes are denoted as bg and bl, respectively, for
the active and passive resonators. Moreover, the passive
mechanical resonator has a weak mechanical Kerr-nonlinearity
denoted by μ. The Hamiltonian describing these coupled
mechanical resonators can be written as

H

=

(Ω0−iΓl)b†lbl+(Ω0+iΓg)b†gbg

(16)

+gmm(b†lbg+blb†g)+μ(b†lbl)2,

where gmm is the coupling strength between the mechanical
modes of the resonators. Generally, the nonlinear Kerr term in
Eq. (16) will shift the
boundary between the PT symmetric regime and the
broken-PT regime. However, in our model, the Kerr
nonlinearity denoted by μ is very weak, and we can omit the
nonlinearity-induced shift of this boundary. To find the boundary
of PT transition, we consider the first three terms in
Eq. (16)

as the mechanical supermodes formed by the coupling of the
resonators. These supermodes b± are characterized by the
eigenfrequencies Ω± and damping rates Γ±.

Figure 7: (Color online). Evolution of the
eigenfrequencies of the coupled mechanical resonators. (a)
Difference of the real parts of the eigenfrequencies of the
supermodes: mode splitting, and (b) difference of the imaginary
parts of the eigenfrequencies (i.e., linewidth) of the supermodes.
The resonance frequencies of the supermodes are non-degenerate in
the PT-symmetric regime. In the broken-PT
symmetry regime, however, they are frequency
degenerate.

For this mechanical PT symmetric system, there are two
different regimes (see Fig. 7):

(i) PT symmetric regime where

Γ=(Γl+Γg)2≤gmm,

(22)

and the two supermodes b+ and b− are nondegenerate in their
resonance frequencies (i.e., real part of their complex
eigenfrequencies) given by

Ω±=Ω0±β=Ω0±√g2mm−Γ2.

(23)

The damping rates of the supermodes (i.e., linewidths of the
resonances; imaginary part of their complex eigenfrequencies) are
the same and equal to

Γ±=χ=Γl−Γg2.

(24)

(ii) Broken PT-symmetry regime where

Γ=Γl+Γg2>gmm.

(25)

The two supermodes b+ and b− are degenerate in their
resonance frequencies

Ω±=Ω0,

(26)

and their damping rates are different:

Γ±=χ∓iβ.

(27)

Now let us consider the nonlinear Kerr term in Eq. (16). Using Eq. (21), we
find

bl

=

√(Ω+−Ω0)2+(Γ+−Γl)2+g2mm(Ω+−Ω−)−i(Γ+−Γ−)b+

−√(Ω−−Ω0)2+(Γ−−Γl)2+g2mm(Ω+−Ω−)−i(Γ+−Γ−)b−

=

βl+b++βl−b−.

By substituting the above equation into the last term on the right
hand side of Eq. (16) and
dropping the non-resonant terms, we can rewrite the nonlinear Kerr
term of Eq. (16) as

Hnl=(|βl+|2b†+b++|βl−|2b†−b−)2.

(28)

The self-Kerr terms
|βl+|4(b†+b+)2 and
|βl−|4(b†−b−)2 only lead to a
frequency-shift of the two supermodes and thus are less important.
The cross-Kerr term

H′nl=|βl+|2|βl−|2(b†+b+)(b†−b−)=μ′(b†+b+)(b†−b−)

(29)

is more important and leads to the redistribution of the nonlinear
effect among the two supermodes. From Eqs. (22)-(24), the
nonlinear coefficient 2|βl+|2|βl−|2 can be
represented in the broken-PT regime as μ′b, and
in the PT symmetric regime as μ′s

μ′b=μΓ2g2mm(Γ2−g2mm)2,μ′s=μg4mm(Γ2−g2mm)2.

(30)

As was observed in photonic
experiments (62); (79), in the
broken-PT regime the two supermodes b± are
degenerate and the field is localized in the gain resonator, and
thus the field bl is much smaller than bg. Therefore, we can
omit the terms related to bl in the expressions of the
supermodes b± and we have

b+≈gmm√(Ω+−Ω0)2+(Γ+−Γl)2+g2mmbg,

b−≈gmm√(Ω−−Ω0)2+(Γ−−Γl)2+g2mmbg.

Subsequently, we find that the cross-Kerr term given in
Eq. (29) can induce a self-Kerr effect in
the gain resonator

H′nl=μg4mm4(Γ2−g2mm)2(b†gbg)2.

(31)

Clearly, when Γ≈gmm (in the vicinity of the
spontaneous PT-symmetry breaking point: the
PT-phase transition point), this self-Kerr
nonlinearity is greatly enhanced.

Appendix B Unidirectional phonon transport by mechanical
nonlinearity

Let us now present a detailed analysis
for finding the unidirectional phonon transport region near the
PT-transition point. In this case, the gain-loss
balance between the active resonator, with annihilation operator
bg, and the passive resonator, with annihilation operator
bl, decreases the effective damping rates of the two modes. In
the vicinity of the PT-phase transition point (i.e.,
Γ≈gmm), the effective damping rates of the two
modes is given by χ=(Γl−Γg)/2. The
coupling between the two mechanical resonators also leads to the
transfer of mechanical Kerr nonlinearity from the passive
resonator to the active resonator, and this mechanical
nonlinearity is strongly enhanced near the
PT-transition point (i.e., Γ≈gmm).
Hereafter, we will denote this enhanced mechanical Kerr
nonlinearity coefficient as μ′b.

Let us first consider the phonon transport from the passive
resonator to the active resonator. Here the phononic field in the
passive resonator is generated via an phononic input field with
strength εd and frequency Ωd. Using the
standard input-output
formalism (81); (80), the output field of
the active mechanical resonator is found as bout=χ1/2bg, which shows that the output field is
proportional to the intracavity field bg, if we omit the vacuum
fluctuations in the input field. Thus the transmission from
passive to active resonator is given by

Tl→g(δ)=χng/|εd|2,

(32)

where ng
represents the steady-state value of the intracavity phonon number
in the active resonator. From the steady-state solution of the
equations of motion for the coupled mechanical resonator system,
we find that ng satisfies

~μ2n3g−2~μ~Ωn2g+(~Γ2+~Ω2)ng−~nin=0,

(33)

where

~Γ=(χ2+δ2+g2mm)χ,~Ω=(χ2+δ2)Ω0−g2mmδ,

~μ=(χ2+δ2)μ′b,~nin=|εd|2g2mm(χ2+δ2).

The algebraic equation (33) has three
or one root depending on the system parameters, and one of the
roots is unstable if the algebraic equation (33) has three roots. When we increase the detuning
δ=Ω0−Ωd, such that

Missing or unrecognized delimiter for \right

(34)

or equivalently,

δ=δmax=g2mm+√g4mm+4(√3χ3−χ2Ω0+√3g2mmχ)(Ω0−√3χ)2(Ω0−√3χ),

(35)

the system enters the bistable regime. In fact, when
δ≤δmax, the algebraic equation has three
branches of solutions. However, two branches of solutions
disappear when δ>δmin (see
Ref. (82) and the supplementary materials
of Ref. (83)). In this case, the transmittance of the
photon transport Tl→g(δ) changes
suddenly from a high value to a low value. Noting that
gmm≫χ near the PT breaking point, the
critical detuning δmax can be approximately
estimated to be

δmax=g2mmΩ0−√3χ.

(36)

Let us now consider the phonon transport from the active
mechanical resonator to the passive one. The driving field with
strength εd and frequency Ωd is fed into the
gain resonator in this case. Following the same discussion and
approach as for the previous case, it can be shown that a
bistability-induced phase transition occurs when the detuning
δ=Ω0−Ωd satisfies

δ−(g2mmΩ0)/(χ2+Ω20)χ(χ2+Ω20+g2mm)/(χ2+Ω20)=√3,

(37)

or equivalently,

δ=δmin=√3(χ2+Ω20+g2mm)χ+g2mmΩ0χ2+Ω20.

(38)

Near the PT-breaking point, χ≪gmm,Ω0, and thus δmin can be
approximately estimated to be

δmin=g2mmΩ0χ2+Ω20.

(39)

Combing Eqs. (36) and (39), we find that when the detuning δ
is within the following region

[δmin,δmax]=[g2mmΩ0Ω20+χ2,g2mmΩ0−√3χ],

(40)

it is possible to observe the unidirectional phonon transport,
i.e., the phonon transport from the passive resonator to the
active resonator is allowed, whereas the phonon transport from the
active resonator to the passive resonator is blocked.

Figure. 8a shows
the transmittance functions Tl→g(δ) and Tg→l(δ)
as a function of the detuning δ. It is (as explained in the
main text) clear that there is a unidirectional phonon transport
region when the detuning is up-scanned from smaller to larger
detuning. We also show in Fig. 8b the rectification ratios for up-scanning and
down-scanning the detuning δ. Similar to our previous
discussions, a non-reciprocal region can be observed for the
up-scanning process, while it disappears for the down-scanning
process, and a high rectification-ratio, larger than 30 dB, can
be obtained within the nonreciprocal region.

Figure 8: (Color online). Bistability curves
and unidirectional phonon-transport regions. (a) Transmittances as
a function of the detuning frequency δ, when the input
field amplitude is fixed at εd. (b) Rectification
ratio for the bidirectional phonon transport versus the detuning
δ: a rectification ratio larger than 30 dB can be
obtained when the detuning is up-scanned to enter the
unidirectional phonon transport region. (c) Transmittances as
function of the intensity of the input field when the detuning
frequency δ is fixed and its value is taken within the
unidirectional phonon transport region in (a). The blue and red
curves represent the power transmittances from the passive to the
active resonator Tl→g and from the active to the
passive resonator Tg→l. The solid and dashed parts
on each curve denote the stable and unstable solutions of the
bistable system. The unstable solutions cannot be observed in the
output and thus lead to sudden transitions (black solid arrows) in
the transmittance functions. (d) Rectification ratios versus
normalized amplitude of the input field for fixed detuning
δ. The melon-colored shaded areas denote the unidirectional
transport regions.

Up to this point, we do not consider the amplitude of the input
field. Let us assume that the detuning δ is fixed and is
within the detuning region given by Eq. (40). We then vary the amplitude of the input
field to show the bistability and the hysteresis in the
transmittance functions. Let us first assume that δ>gmm.
If we consider the phonon transport from the passive resonator to
the active resonator, we can obtain an algebraic equation similar
to that given in Eq. (33). The
bistable transition point corresponds to the stationary points of
the function

f(ng)=~μ2n3g−2~μ~Ωn2g+(~Γ2+~Ω2)ng.

(41)

By setting f′(ng)=0, the stationary point of f(ng) can be
found as

n∗g=[2~μ~Ω−√4~μ2~Ω2−3~μ2(~Γ2+~Ω2)](3~μ2)−1.

(42)

The upper bound of the unidirectional phonon transport region is
given by

Near the PT-transition point, we have
δ,gmm≫χ, and thus it can be approximately
estimated that

|εmax|2≈2(δ2+g2mm)29μ′bg2mmδ.

(43)

Let us now consider the case of phonon transport from the active
resonator to the passive resonator when the amplitude of the input
field is varied and the detuning is kept fixed. In this case, we
obtain

~~μ2n3g−2~~μ~~Ωn2g+(~~Γ2+~~Ω2)ng−~~nin=0,

(44)

where

~~Γ=(χ2+Ω20+g2mm)χ,~~Ω=(χ2+Ω2)δ−g2mmΩ0,

~~μ=(χ2+Ω20)μ′b,~~nin=(χ2+δ2)2|εd|2.

Similar to Eq. (41), the bistable transition point can be found by
calculating the stationary points of the function

f(ng)=~~μ2n3g−2~~μ~~Ωn2g+(~~Γ2+~~Ω2)ng.

(45)

which leads to

~n∗g=[2~~μ~~Ω−√4~~μ2~~Ω2−3~~μ2(~~Γ2+~~Ω2)](3~~μ2)−1.

(46)

The lower bound of the unidirectional phonon transport region is
then given by

Near the PT-transition point, we have
δ,gmm≫χ, and it can be approximately estimated
that

|εmin|2≈2(δ2+g2mm)29μ′bδ3.

(47)

We thus conclude that nonreciprocal phonon transport takes place
if the amplitude of the input is within the region

|εd|2∈⎡⎣2(δ2+g2mm)39μ′bδ3,2(δ2+g2mm)39μ′bg2mmδ⎤⎦.

(48)

Similarly, when δ≤gmm, the nonreciprocal region for
the amplitude of the input field can be written as

|εd|2∈⎡⎣2(δ2+g2mm)39μ′bg2mmδ,2(δ2+g2mm)39μ′bδ3⎤⎦.

(49)

In Fig. 8c, we
present the transmittances as a function of the amplitude of the
input field when the detuning is kept fixed within the
unidirectional transport region given in Eq. (40). We see that the lower stable branches
of the bistable curves shown in Fig. 8c (the parts of the bistable curves before
the bistable transitions occur) increase when we increase the
intensity of the input field. This decreases the rectification, as
shown in Fig. 8d.

Appendix C Can this system be used as a phonon isolator?

In order to check the performance of the proposed system as an
isolator for phonons, we study the system considering that phonons
are injected in the system in both directions, that is
simultaneously at the passive and active resonator sides. If the
system exhibits unidirectional phonon transport under this
condition, then the proposed system can be used as an isolator.

Figure 9: (Color online). Bistability curves for
the mechanical PT system when phonons are input
simultaneously in both directions. (a) Transmittances as functions
of the detuning frequency δ when the input field amplitude
is fixed at εd. (b) Transmittances as functions of
the amplitude of the input field when the detuning frequency
δ is fixed. The blue and red curves represent the power
transmittance functions Tl→g and Tg→l. The solid and dashed parts on each curve denote the stable
and unstable solutions of the bistable systems. The unstable
solutions cannot be observed in the output and thus lead to sudden
transitions in the transmittance functions.

The equations of motion of the system for this case can be written
as

˙bl=−(χ+iδl)bl−igmmbg+iεl,

˙bg=−(χ+iδg)bg−iμ′b(b†gbg)bg−igmmbl+iεg,

where the last terms on the right-hand-sides of
Eq. (C) denote the
input fields. The steady-state solution of Eq. (C) leads to

¯μ2n3g−2¯μ¯Ωn2g+(¯Γ2+¯Ω2)ng−¯nin=0,

(51)

where

¯Γ=(χ2+δ2l+g2mm)χ,¯μ=(χ2+δ2l)μ′b

¯Ω=(χ2+δ2l)δg−g2mmδl,

¯nin=|εl|2g2mm(χ2+δ2l)+(χ2+δ2l)2|εg|2.

Let us first fix εl,εg,δg, and
vary the detuning δl=δ. In this case, the bistable
transitions for both directions occur when the detuning δ
satisfies

Missing or unrecognized delimiter for \right

(52)

When the detuning is up-scanned from smaller to larger detuning
values, the bistable transition occurs for

δ

=

Missing or unrecognized delimiter for \right

(53)

+g2mm2(δg−√3χ).

When the detuning δ is down-scanned from larger to smaller
detuning values, the bistable transition occur at

δ

=

−


⎷g4mm4(δg−√3χ)2+(√3χ3−χ2δg+√3g2mmχ)(δg−√3χ)

(54)

+g2mm2(δg−√3χ).

The transmittances presented in Fig. 9a clearly show the bistable operation. A close
look at Fig. 9a
reveals that the transition from the bistable region to the stable
trajectories takes place at the same points for both directions.
We cannot find a detuning region within which transport in one
direction is allowed and the transport in the other direction is
prevented. Thus, we conclude that when phonons are injected
simultaneously at both input ports, we cannot see a unidirectional
operation. Consequently, it is impossible to use this system as an
isolator for phonons.

Let us now fix δl, δg, εg, and vary
εl=εd, to check the possibility of
providing a phonon isolator. The bistable transition point is just
the stationary points of the function

¯f(ng)=¯μ2n3g−2¯μ¯Ωn2g+(¯Γ2+¯Ω2)ng−|εg|2.

(55)

By setting ¯f′(ng)=0, we find

¯n∗g1

=

[2¯μ¯Ω+√4¯μ2¯Ω2−3¯μ2(¯Γ2+¯Ω2)](3¯μ2)−1,

¯n∗g2

=

[2¯μ¯Ω−√4¯μ2¯Ω2−3¯μ2(¯Γ2+¯Ω2)](3¯μ2)−1,

The bistable transition occurs at

|εd|2=f(n∗g1)g2mm(χ2+δ2l)

(56)

when the amplitude of the input field εd is
up-scanned and for

|εd|2=f(n∗g2)g2mm(χ2+δ2l)

(57)

when the amplitude of the input field is down-scanned (see
Fig. 9). For this
case too, we do not see a unidirectional phonon transport region
if we feed the inputs at the active and passive resonators sides
simultaneously. Thus we conclude that although the proposed system
can be used as phonon diode allowing nonreciprocal phonon
transport, it cannot function as an isolator for phonons.